16. A proportion is said to be multiplied by a number, when it is so often taken as there be unities in that number; and if the proportion be of the greater to the less, then shall also the quantity of the proportion be increased by the multiplication; but when the proportion is of the less to the greater, then as the number increaseth, the quantity of the proportion diminisheth; as in these three numbers, 4, 2, 1, the proportion of 4 to 1 is not only the duplicate of 4 to 2, but also twice as great; but inverting the order of those numbers thus, 1, 2, 4, the proportion of 1 to 2 is greater than that of 1 to 4; and therefore though the proportion of 1 to 4 be the duplicate of 1 to 2, yet it is not twice so great as that of 1 to 2, but contrarily the half of it. In like manner, a proportion is said to be divided, when between two quantities are interposed one or more means in continual proportion, and then the proportion of the first to the second is said to be subduplicate of that of the first to the third, and subtriplicate of that of the first to the fourth, &c.

This mixture of proportions, where some are proportions of excess, others of defect, as in a merchant's account of debtor and creditor, is not so easily reckoned as some think; but maketh the composition of proportions sometimes to be addition, sometimes substraction; which soundeth absurdly to such as have always by composition understood addition, and by diminution substraction. Therefore to make this account a little clearer, we are to consider (that which is commonly assumed, and truly) that if there be never so many quantities, the proportion of the first to the last is compounded of the proportions of the first to the second, and of the second to the third, and so on to the last, without regarding their equality, excess, or defect; so that if two proportions, one of inequality, the other of equality, be added together, the proportion is not thereby made greater nor less; as for example, if the proportions of A to B and of B to B be compounded, the proportion of the first to the second is as much as the sum of both, because proportion of equality, being not quantity, neither augmenteth quantity nor lesseneth it. But if there be three quantities, A, B, C, unequal, and the first be the greatest, the last least, then the proportion of B to C is an addition to that of A to B, and makes it greater; and on the contrary, if A be the least, and C the greatest quantity, then doth the addition of the proportion of B to C make the compounded proportion of A to C less than the proportion of A to B, that is, the whole less than the part. The composition therefore of proportions is not in this case the augmentation of them, but the diminution; for the same quantity (Euclid V. 8) compared with two other quantities, hath a greater proportion to the lesser of them than to the greater. Likewise, when the proportions compounded are one of excess, the other of defect, if the first be of excess, as in these numbers, 8, 6, 9, the proportion compounded, namely, of 8 to 9, is less than the proportion of one of the parts of it, namely, of 8 to 6; but if the proportion of the first to the second be of defect, and that of the second to the third be of excess, as in these numbers, 6, 8, 4, then shall the proportion of the first to the third be greater than that of the first to the second, as 6 hath a greater proportion to 4 than to 8; the reason whereof is manifestly this, that the less any quantity is deficient of another, or the more one exceedeth another, the proportion of it to that other is the greater. Suppose now three quantities in continual proportion, A B 4, A C 6, A D 9. Because therefore A D is greater than A C, but not greater than A D, the proportion of A D to AC will be (by Euclid, V. 8) greater than that of A D to A D; and likewise, because the proportions of A D to A C, and of A C to A B are the same, the proportions of A D to A C and of A C to A B, being both proportions of excess, make the whole proportion of A D to A B, or of 9 to 4, not only the duplicate of A D to A C, that is, of 9 to 6, but also the double, or twice so great. On the other side, because the proportion of A D to A D, or 9 to 9, being proportion of equality, is no quantity, and yet greater than that of A C to A D, or 6 to 9, it will be as 0 - 9 to 0 - 6, so A C to A D, and again, as 0 - 9 to 0 - 6, so 0 - 6 to 0 - 4; but 0 - 4, 0 - 6, 0 - 9 are in continual proportion; and because 0 - 4 is greater than 0 - 6, the proportion of 0 - 4 to 0 - 6 will be double to the proportion of 0 - 4 to 0 - 9, double I say, and yet not duplicate, but subduplicate.

If any be unsatisfied with this ratiocination, let him first consider that (by Euclid V. 8) the proportion of A B to A C is greater than that of A B to A D, wheresoever D be |  Ḅ  C̣   Ḍ
A——————E| placed in the line A C prolonged; and the further off the point D is from C, so much the greater is the proportion of A B to A C than that of A B to A D. There is therefore some point (which suppose be E) in such distance from C, as that the proportion of A B to A C will be twice as great as that of A B to A E. That considered, let him determine the length of the line A E, and demonstrate, if he can, that A E is greater or less than A D.

By the same method, if there be more quantities than three, as A, B, C, D, in continual proportion, and A be the least, it may be made appear that the proportion of A to B is triple magnitude, though subtriple in multitude, to the proportion of A to D.

17. If there be never so many quantities, the number whereof is odd, and their order such, that from the middlemost quantity both ways they proceed in continual proportion, the proportion of the two which are next on either side to the middlemost is subduplicate to the proportion of the two which are next to these on both sides, and subtriplicate of the proportion of the two which are yet one place more remote, &c. For let the magnitudes be C, B, A, D, E, and let A, B, C, as also A, D, E be in continual proportion; I say the proportion of D to B is subduplicate of the proportion of E to C. For the proportion of D to B is compounded of the proportions of D to A, and of A to B once taken; but the proportion of E to C is compounded of the same twice taken; and therefore the proportion of D to B is subduplicate of the proportion of E to C. And in the same manner, if there were three terms on either side, it might be demonstrated that the proportion of D to B would be subtriplicate of that of the extremes, &c.

18. If there be never so many continual proportionals, as the first, second, third, &c. their differences will be proportional to them. For the second, third, &c. are severally consequents of the preceding, and antecedents of the following proportion. But (by art. 10) the difference of the first antecedent and consequent, to difference of the second antecedent and consequent, is as the first antecedent to the second antecedent, that is, as the first term to the second, or as the second to the third, &c. in continual proportionals.

19. If there be three continual proportionals, the sum of the extremes, together with the mean twice taken, the sum of the mean and either of the extremes, and the same extreme, are continual proportionals. For let A. B. C be continual proportionals. Seeing, therefore, A. B :: B. C are proportionals, by composition also A + B. B :: B + C. C will be proportionals; and by permutation A + B. B + C :: B. C will also be proportionals; and again, by composition A + 2B + C. B + C :: B + C. C; which was to be proved.

20. In four continual proportionals, the greatest and the least put together is a greater quantity than the other two put together. Let A. B :: C. D be continual proportionals; whereof let the greatest be A, and the least be D; I say A + D is greater than B + C. For by art. 10, A - B. C - D :: A. C are proportionals; and therefore A - B is, by art. 11, greater than C - D. Add B on both sides, and A will be greater than C + B - D. And again, add D on both sides, and A + D will be greater than B + C; which was to be proved.

21. If there be four proportionals, the extremes multiplied into one another, and the means multiplied into one another, will make equal products. Let A. B :: C. D be proportionals; I say A D is equal to B C. For the proportion of A D to B C is compounded, by art. 13, of the proportions of A to B, and D to C, that is, its inverse B to A; and therefore, by art. 14, this compounded proportion is the proportion of equality; and therefore also, the proportion of A D to B C is the proportion of equality. Wherefore they are equal.

22. If there be four quantities, and the proportion of the first to the second be duplicate of the proportion of the third to the fourth, the product of the extremes to the product of the means, will be as the third to the fourth. Let the four quantities be A, B, C and D; and let the proportion of A to B be duplicate of the proportion of C to D, I say A D, that is, the product of A into D is to B C, that is, to the product of the means, as C to D. For seeing the proportion of A to B is duplicate of the proportion of C to D, if it be as C to D, so D to another, E, then A. B :: C. E will be proportionals; for the proportion of A to B is by supposition duplicate of the proportion of C to D; and C to E duplicate also of that of C to D by the definition, art. 15. Wherefore, by the last article, A E or A into E is equal to B C or B into C; but, by coroll. IV. art. 6, A D is to A E as D to E, that is, as C to D; and therefore A D is to B C, which as I have shown is equal to A E, as C to D; which was to be proved.

Moreover, if the proportion of the first A to the second B be triplicate of the proportion of the third C to the fourth D, the product of the extremes to the product of the means will be duplicate of the proportion of the third to the fourth. For if it be as C to D so D to E, and again, as D to E so E to another, F, then the proportion of C to F will be triplicate of the proportion of C to D; and consequently, A. B :: C. F will be proportionals, and A F equal to B C. But as A D to A F, so is D to F; and therefore, also, as A D to B C, so D to F, that is, so C to E; but the proportion of C to E is duplicate of the proportion of C to D; wherefore, also, the proportion of A D to B C is duplicate of that of C to D, as was propounded.

23. If there be four proportionals, and a mean be interposed betwixt the first and second, and another betwixt the third and fourth, the first of these means will be to the second, as the first of the proportionals is to the third, or as the second of them is to the fourth. For let A. B :: C. D be proportionals, and let E be a mean betwixt A and B, and F a mean betwixt C and D; I say A. C :: E. F are proportionals. For the proportion of A to E is subduplicate of the proportion of A to B, or of C to D. Also, the proportion of C to F is subduplicate of that of C to D; and therefore A. E :: C. F are proportionals; and by permutation A. C :: E. F are also proportionals; which was to be proved.

24. Any thing is said to be divided into extreme and mean proportion, when the whole and the parts are in continual proportion. As for example, when A + B. A. B are continual proportionals; or when the straight line A C is so divided in B, that A C. A B. B C are in continual proportion. And if the same line A C be again divided |A       B   C
——+——+——
    D| in D, so as that A C. C D. A D be continual proportionals; then also A C. A B. A D will be continual proportionals; and in like manner, though in contrary order, C A. C D. C B will be continual proportionals; which cannot happen in any line otherwise divided.

25. If there be three continual proportionals, and again, three other continual proportions, which have the same middle term, their extremes will be in reciprocal proportion. For let A. B. C and D. B. E be continual proportionals, I say A. D :: E. C shall be proportionals. For the proportion of A to D is compounded of the proportions of A to B, and of B to D; and the proportion of E to C is compounded of those of E to B, that is, of B to D, and of B to C, that is, of A to B. Wherefore, by equality, A. D :: E. C are proportionals.

26. If any two unequal quantities be made extremes, and there be interposed betwixt them any number of means in geometrical proportion, and the same number of means in arithmetical proportion, the several means in geometrical proportion will be less than the several means in arithmetical proportion. For betwixt A the lesser, and E the greater extreme, let there be interposed three means, B, C, D, in geometrical proportion, and as many more, F, G, H, in arithmetical proportion; I say B will be less than F, C than G, and D than H. For first, the difference between A and F is the same with that between F and G, and with that between G and H, by the definition of arithmetical proportion; and therefore, the difference of the proportionals which stand next to one another, to the difference of the extremes, is, when there is but one mean, half their difference; when two, a third part of it; when three, a quarter, &c.; so that in this example it is a quarter. But the difference between D and E, by art. 17, is more than a |   A      A
   -       -
   B      F
   —     —
   C      G
  –—    –—
   D     H
  ——   ——
   E      E
 ——— ———| quarter of the difference between the extremes, because the proportion is geometrical, and therefore the difference between A and D is less than three quarters of the same difference of the extremes. In like manner, if the difference between A and D be understood to be divided into three equal parts, it may be proved, that the difference between A and C is less than two quarters of the difference of the extremes A and E. And lastly, if the difference between A and C be divided into two equal parts, that the difference between A and B is less than a quarter of the difference of the extremes A and E.

From the consideration hereof, it is manifest, that B, that is A together with something else which is less than a fourth part of the difference of the extremes A and E, is less than F, that is, than the same A with something else which is equal to the said fourth part. Also, that C, that is A with something else which is less than two fourth parts of the said difference, is less than G, that is, than A together with the said two-fourths. And lastly, that D, which exceeds A by less than three-fourths of the said difference, is less than H, which exceeds the same A by three entire fourths of the said difference. And in the same manner it would be if there were four means, saving that instead of fourths of the difference of the extremes we are to take fifth parts; and so on.

27. Lemma. If a quantity being given, first one quantity be both added to it and subtracted from it, and then another greater or less, the proportion of the remainder to the aggregate, is greater where the less quantity is added and substracted, than where the greater quantity is added and substracted. Let B be added to and substracted from the quantity A; so that A - B be the remainder, and A + B the aggregate; and again, let C, a greater quantity than B, be added to and substracted from the same A, so that A - C be the remainder and A + C the aggregate; I say A - B. A + B :: A - C. A + C will be an hyperlogism. For A - B. A :: A - C. A is an hyperlogism of a greater antecedent to the same consequent; and therefore A - B. A + B :: A - C. A + C is a much greater hyperlogism, being made of a greater antecedent to a less consequent.

28. If unequal parts be taken from two equal quantities, and betwixt the whole and the part of each there be interposed two means, one in geometrical, the other in arithmetical proportion; the difference betwixt the two means will be greatest, where the difference betwixt the whole and its part is greatest. |A   E   G   H   B
───┼──┼──┼───
────┼──┼──┼──
A    F    I   K  B|For let A B and A B be two equal quantities, from which let two unequal parts be taken, namely, A E the less, and A F the greater; and betwixt A B and A E let A G be a mean in geometrical proportion, and A H a mean in arithmetical proportion. Also betwixt A B and A F let A I be a mean in geometrical proportion, and A K a mean in arithmetical proportion; I say H G is greater than K I.

Now seeing the proportion of A B to A E is greater than that of A B to A F, the proportion of A B to A G, which is half the greater proportion, is greater than the proportion of A B to A I the half of the less proportion; and therefore A I is greater than A G. Wherefore the proportion of A B - A G to A B + A G, by the precedent lemma, will be greater than the proportion of A B - A I to A B + A I; and therefore also the proportion of H G to B H will be greater than that of K I to B K, and much greater than the proportion of K I to B H, which is greater than B K; for B H is the half of B E, as B K is the half of B F, which, by supposition, is less than B E. Wherefore H G is greater than K I; which was to be proved.

Coroll. It is manifest from hence, that if any quantity be supposed to be divided into equal parts infinite in number, the difference between the arithmetical and geometrical means will be infinitely little, that is, none at all. And upon this foundation, chiefly, the art of making those numbers, which are called Logarithms, seems to have been built.

29. If any number of quantities be propounded, whether they be unequal, or equal to one another; and there be another quantity, which multiplied by the number of the propounded quantities, is equal to them all; that other quantity is a mean in arithmetical proportion to all those propounded quantities.

CHAP. XIV.

OF STRAIT AND CROOKED, ANGLE AND
FIGURE.

1. The definition and properties of a strait line.—2. The definition and properties of a plane superficies.—3. Several sorts of crooked lines.—4. The definition and properties of a circular line.—5. The properties of a strait line taken in a plane.—6.. The definition of tangent lines.—7. The definition of an angle, and the kinds thereof.—8. In concentric circles, arches of the same angle are to one another, as the whole circumferences are.—9. The quantity of an angle, in what it consists. —10. The distinction of angles, simply so called.—11. Of strait lines from the centre of a circle to a tangent of the same. —12. The general definition of parallels, and the properties of strait parallels.—13. The circumferences of circles are to one another, as their diameters are.—14. In triangles, strait lines parallel to the bases are to one another, as the parts of the sides which they cut off from the vertex.—15. By what fraction of a strait line the circumference of a circle is made.—16. That an angle of contingence is quantity, but of a different kind from that of an angle simply so called; and that it can neither add nor take away any thing from the same.—17. That the inclination of planes is angle simply so called.—18. A solid angle what it is.—19. What is the nature of asymptotes.—20. Situation, by what it is determined.—21. What is like situation; what is figure; and what are like figures.

1. Between two points given, the shortest line is that, whose extreme points cannot be drawn farther asunder without altering the quantity, that is, without altering the proportion of that line to any other line given. For the magnitude of a line is computed by the greatest distance which may be between its extreme points; so that any one line, whether it be extended or bowed, has always one and the same length, because it can have but one greatest distance between its extreme points.

And seeing the action, by which a strait line is made crooked, or contrarily a crooked line is made strait, is nothing but the bringing of its extreme points nearer to one another, or the setting of them further asunder, a crooked line may rightly be defined to be that, whose extreme points may be understood to be drawn further asunder; and a strait line to be that, whose extreme points cannot be drawn further asunder; and comparatively, a more crooked, to be that line whose extreme points are nearer to one another than those of the other, supposing both the lines to be of equal length. Now, howsoever a line be bowed, it makes always a sinus or cavity, sometimes on one side, sometimes on another; so that the same crooked line may either have its whole cavity on one side only, or it may have it part on one side and part on the other side. Which being well understood, it will be easy to understand the following comparisons of strait and crooked lines.

First, if a strait and a crooked line have their extreme points common, the crooked line is longer than the strait line. For if the extreme points of the crooked line be drawn out to their greatest distance, it will be made a strait line, of which that, which was a strait line from the beginning, will be but a part; and therefore the strait line was shorter than the crooked line, which had the same extreme points. And for the same reason, if two crooked lines have their extreme points common, and both of them have all their cavity on one and the same side, the outermost of the two will be the longest line.

Secondly, a strait line and a perpetually crooked line cannot be coincident, no, not in the least part. For if they should, then not only some strait line would have its extreme points common with some crooked line, but also they would, by reason of their coincidence, be equal to one another; which, as I have newly shown, cannot be.

Thirdly, between two points given, there can be understood but one strait line; because there cannot be more than one least interval or length between the same points. For if there may be two, they will either be coincident, and so both of them will be one strait line; or if they be not coincident, then the application of one to the other by extension will make the extended line have its extreme points at greater distance than the other; and consequently, it was crooked from the beginning.

Fourthly, from this last it follows, that two strait lines cannot include a superficies. For if they have both their extreme points common, they are coincident; and if they have but one or neither of them common, then at one or both ends the extreme points will be disjoined, and include no superficies, but leave all open and undetermined.

Fifthly, every part of a strait line is a strait line. For seeing every part of a strait line is the least that can be drawn between its own extreme points, if all the parts should not constitute a strait line, they would altogether be longer than the whole line.

2. A plane or a plane superficies, is that which is described by a strait line so moved, that all the several points thereof describe several strait lines. A strait line, therefore, is necessarily all of it in the same plane which it describes. Also the strait lines, which are made by the points that describe a plane, are all of them in the same plane. Moreover, if any line whatsoever be moved in a plane, the lines, which are described by it, are all of them in the same plane.

All other superficies, which are not plane, are crooked, that is, are either concave or convex. And the same comparisons, which were made of strait and crooked lines, may also be made of plane and crooked superficies.

For, first, if a plane and crooked superficies be terminated with the same lines, the crooked superficies is greater than the plane superficies. For if the lines, of which the crooked superficies consists, be extended, they will be found to be longer than those of which the plane superficies consists, which cannot be extended, because they are strait.

Secondly, two superficies, whereof the one is plane, and the other continually crooked, cannot be coincident, no, not in the least part. For if they were coincident, they would be equal; nay, the same superficies would be both plane and crooked, which is impossible.

Thirdly, within the same terminating lines there can be no more than one plane superficies; because there can be but one least superficies within the same.

Fourthly, no number of plane superficies can include a solid, unless more than two of them end in a common vertex. For if two planes have both the same terminating lines, they are coincident, that is, they are but one superficies; and if their terminating lines be not the same, they leave one or more sides open.

Fifthly, every part of a plane superficies is a plane superficies. For seeing the whole plane superficies is the least of all those, that have the same terminating lines; and also every part of the same superficies is the least of all those, that are terminated with the same lines; if every part should not constitute a plane superficies, all the parts put together would not be equal to the whole.

3. Of straitness, whether it be in lines or in superficies, there is but one kind; but of crookedness there are many kinds; for of crooked magnitudes, some are congruous, that is, are coincident when they are applied to one other; others are incongruous. Again, some are ὁμοιομερεῖς or uniform, that is, have their parts, howsoever taken, congruous to one another; others are ἀνομοιομερεῖς or of several forms. Moreover, of such as are crooked, some are continually crooked, others have parts which are not crooked.

4. If a strait line be moved in a plane, in such manner, that while one end of it stands still, the whole line be carried round about till it come again into the same place from whence it was first moved, it will describe a plane superficies, which will be terminated every way by that crooked line, which is made by that end of the strait line which was carried round. Now this superficies is called a CIRCLE; and of this circle, the unmoved point is the centre; the crooked line which terminates it, the perimeter; and every part of that crooked line, a circumference or arch; the strait line, which generated the circle, is the semidiameter or radius; and any strait line, which passeth through the centre and is terminated on both sides in the circumference, is called the diameter. Moreover, every point of the radius, which describes the circle, describes in the same time its own perimeter, terminating its own circle, which is said to be concentric to all the other circles, because this and all those have one common centre.

Wherefore in every circle, all strait lines from the centre to the circumference are equal. For they are all coincident with the radius which generates the circle.

Also the diameter divides both the perimeter and the circle itself into two equal parts. For if those two parts be applied to one another, and the semiperimeters be coincident, then, seeing they have one common diameter, they will be equal; and the semicircles will be equal also; for these also will be coincident. But if the semiperimeters be not coincident, then some one strait line, which passes through the centre, which centre is in the diameter, will be cut by them in two points. Wherefore, seeing all the strait lines from the centre to the circumference are equal, a part of the same strait line will be equal to the whole; which is impossible.

For the same reason the perimeter of a circle will be uniform, that is, any one part of it will be coincident with any other equal part of the same.

5. From hence may be collected this property of a strait line, namely, that it is all contained in that plane which contains both its extreme points. For seeing both its extreme points are in the plane, that strait line, which describes the plane, will pass through them both; and if one of them be made a centre, and at the distance between both a circumference be described, whose radius is the strait line which describes the plane, that circumference will pass through the other point. Wherefore between the two propounded points, there is one strait line, by the definition of a circle, contained wholly in the propounded plane; and therefore if another strait line might be drawn between the same points, and yet not be contained in the same plane, it would follow, that between two points two strait lines may be drawn; which has been demonstrated to be impossible.

It may also be collected, that if two planes cut one another, their common section will be a strait line. For the two extreme points of the intersection are in both the intersecting planes; and between those points a strait line may be drawn; but a strait line between any two points is in the same plane, in which the points are; and seeing these are in both the planes, the strait line which connects them will also be in both the same planes, and therefore it is the common section of both. And every other line, that can be drawn between those points, will be either coincident with that line, that is, it will be the same line; or it will not be coincident, and then it will be in neither, or but in one of those planes.

As a strait line may be understood to be moved round about whilst one end thereof remains fixed, as the centre; so in like manner it is easy to understand, that a plane may be circumduced about a strait line, whilst the strait line remains still in one and the same place, as the axis of that motion. Now from hence it is manifest, that any three points are in some one plane. For as any two points, if they be connected by a strait line, are understood to be in the same plane in which the strait line is; so, if that plane be circumduced about the same strait line, it will in its revolution take in any third point, howsoever it be situate; and then the three points will be all in that plane; and consequently the three strait lines which connect those points, will also be in the same plane.

6. Two lines are said to touch one another, which being both drawn to one and the same point, will not cut one another, though they be produced, produced, I say, in the same manner in which they were generated. And therefore if two strait lines touch one another in any one point, they will be contiguous through their whole length. Also two lines continually crooked will do the same, if they be congruous and be applied to one another according to their congruity; otherwise, if they be incongruously applied, they will, as all other crooked lines, touch one another, where they touch, but in one point only. Which is manifest from this, that there can be no congruity between a strait line and a line that is continually crooked; for otherwise the same line might be both strait and crooked. Besides, when a strait line touches a crooked line, if the strait line be never so little moved about upon the point of contact, it will cut the crooked line; for seeing it touches it but in one point, if it incline any way, it will do more than touch it; that is, it will either be congruous to it, or it will cut it; but it cannot be congruous to it; and therefore it will cut it.

7. An angle, according to the most general acceptation of the word, may be thus defined; when two lines, or many superficies, concur in one sole point, and diverge every where else, the quantity of that divergence is an ANGLE. And an angle is of two sorts; for, first, it may be made by the concurrence of lines, and then it is a superficial angle; or by the concurrence of superficies, and then it is called a solid angle.

Again, from the two ways by which two lines may diverge from one another, superficial angles are divided into two kinds. For two strait lines, which are applied to one another, and are contiguous in their whole length, may be separated or pulled open in such manner, that their concurrence in one point will still remain; and this separation or opening may be either by circular motion, the centre whereof is their point of concurrence, and the lines will still retain their straitness, the quantity of which separation or divergence is an angle simply so called; or they may be separated by continual flexion or curvation in every imaginable point; and the quantity of this separation is that, which is called an angle of contingence.

Besides, of superficial angles simply so called, those, which are in a plane superficies, are plane; and those, which are not plane, are denominated from the superficies in which they are.

Lastly, those are strait-lined angles, which are made by strait lines; as those which are made by crooked lines are crooked-lined; and those which are made both of strait and crooked lines, are mixed angles.

8. Two arches intercepted between two radii of concentric circles, have the same proportion to one another, which their whole perimeters have to one another. For let the point A (in the first figure) be the centre of the two circles B C D and E F G, in which the radii A E B and A F C intercept the arches B C and E F; I say the proportion of the arch B C to the arch E F is the same with that of the perimeter B C D to the perimeter E F G. For if the radius A F C be understood to be moved about the centre A with circular and uniform motion, that is, with equal swiftness everywhere, the point C will in a certain time describe the perimeter B C D, and in a part of that time the arch B C; and because the velocities are equal by which both the arch and the whole perimeter are described, the proportion of the magnitude of the perimeter B C D to the magnitude of the arch BC is determined by nothing but the difference of the times in which the perimeter and the arch are described. But both the perimeters are described in one and the same time, and both the arches in one and the same time; and therefore the proportions of the perimeter B C D to the arch B C, and of the perimeter E F G to the arch E F, are both determined by the same cause. Wherefore B C D. B C :: E F G. E F are proportionals (by the 6th art. of the last chapter), and by permutation B C D. E F G :: B C. E F will also be proportionals; which was to be demonstrated.

9. Nothing is contributed towards the quantity of an angle, neither by the length, nor by the equality, nor by the inequality of the lines which comprehend it. For the lines A B and A C comprehend the same angle which is comprehended by the lines A E and A F, or A B and A F. Nor is an angle either increased or diminished by the absolute quantity of the arch, which subtends the same; for both the greater arch B C and the lesser arch E F are subtended to the same angle. But the quantity of an angle is estimated by the quantity of the subtending arch compared with the quantity of the whole perimeter. And therefore the quantity of an angle simply so called may be thus defined: the quantity of an angle is an arch or circumference of a circle, determined by its proportion to the whole perimeter. So that when an arch is intercepted between two strait lines drawn from the centre, look how great a portion that arch is of the whole perimeter, so great is the angle. From whence it may be understood, that when the lines which contain an angle are strait lines, the quantity of that angle may be taken at any distance from the centre. But if one or both of the containing lines be crooked, then the quantity of the angle is to be taken in the least distance from the centre, or from their concurrence; for the least distance is to be considered as a strait line, seeing no crooked line can be imagined so little, but that there may be a less strait line. And although the least strait line cannot be given, because the least given line may still be divided, yet we may come to a part so small, as is not at all considerable; which we call a point. And this point may be understood to be in a strait line which touches a crooked line; for an angle is generated by separating, by circular motion, one strait line from another which touches it, as has been said above in the 7th article. Wherefore an angle, which two crooked lines make, is the same with that which is made by two strait lines which touch them.

10. From hence it follows, that vertical angles, such as are A B C, D B F in the second figure, are equal to one another. For if, from the two semiperimeters D A C, F D A, which are equal to one another, the common arch D A be taken away, the remaining arches A C, D F will be equal to one another.

Another distinction of angles is into right and oblique. A right angle is that, whose quantity is the fourth part of the perimeter. And the lines, which make a right angle, are said to be perpendicular to one another. Also, of oblique angles, that which is greater than a right, is called an obtuse angle; and that which is less, an acute angle. From whence it follows, that all the angles that can possibly be made at one and the same point, together taken, are equal to four right angles; because the quantities of them all put together make the whole perimeter. Also, that all the angles, which are made on one side of a strait line, from any one point taken in the same, are equal to two right angles; for if that point be made the centre, that strait line will be the diameter of a circle, by whose circumference the quantity of an angle is determined; and that diameter will divide the perimeter into two equal parts.